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Chapter 33 & 34 Review. Chapter 33: The Magnetic Field. Cyclotron Motion. Newton’s Law. Both a and F directed toward center of circle. Equate. What is the force on those N electrons?. Force on a single electron. Force on N electrons. Now use.
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Cyclotron Motion Newton’s Law Both a and F directed toward center of circle Equate
What is the force on those N electrons? Force on a single electron Force on N electrons Now use points parallel to the wire in the direction of the current when I is positive Lets make l a vector Force on wire segment:
Comments: Force is perpendicular to both B and l Force is proportional to I, B, and length of line segment Superposition: To find the total force on a wire you must break it into segments and sum up the contributions from each segment
Electric Field Magnetic Field r What are the magnitudes and directions of the electric and magnetic fields at this point? Assume q > 0 q Comparisons: both go like r-2, are proportional to q, have 4p in the denominator, have funny Greek letters Differences: E along r, B perpendicular to r and v
Magnetic Field due to a current Magnetic Field due to a single charge If many charges use superposition r3 r2 #3 q3 , v3 Where I want to know what B is r1 #2 q2 , v2 #1 q1, v1
For moving charges in a wire, first sum over charges in each segment, then sum over segments
Summing over segments - integrating along curve Biot Savart law r I Integral expression looks simple but…..you have to keep track of two position vectors which is where you want to know B which is the location of the line segment that is contributing to B. This is what you integrate over.
Magnetic field due to an infinitely long wire z Current I flows along z axis I I want to find B at the point x I will sum over segments at points
r compare with E-field for a line charge
Biot-Savart Law implies Gauss’ Law and Amperes Law Ampere’s Law Gauss’ Law: But also, Gauss’ law and Ampere’s Law imply the Biot -Savart law
Magnetic field due to a single loop of current Electric field due to a single charge Guassian surfaces
Biot-Savart Law implies Gauss’ Law and Amperes Law Ampere’s Law Gauss’ Law: But also, Gauss’ law and Ampere’s Law imply the Biot -Savart law
Magnetic Flux Some surface Remember for a closed surface Closed surface Open surface Magnetic flux measures how much magnetic field passes through a given surface
Rectangular surface in a constant magnetic field. Flux depends on orientation of surface relative to direction of B Suppose the rectangle is oriented do that are parallel
Lenz’s Law In a loop through which there is a change in magnetic flux, and EMF is induced that tends to resist the change in flux What is the direction of the magnetic field made by the current I? Into the page B. Out of the page
Reasons Flux Through a Loop Can Change Location of loop can change Shape of loop can change Orientation of loop can change Magnetic field can change
Faraday’s Law for Moving Loops Faraday’s Law for Stationary Loops
R I Now I have cleaned things up making use of IB=-IR, IR=IL=I. Now use device laws: VR = RI VL = L dI/dt I VB VL VR L KVL: VL + VR - VB = 0 This is a differential equation that determines I(t). Need an initial condition I(0)=0
Approaches a value VB/R Current starts at zero Solution: This is called the “L over R” time. Let’s verify
R I Initially I is small and VR is small. All of VB falls across the inductor, VL=VB. Inductor acts like an open circuit. I VB VL VR L Time asymptotically I stops changing and VL is small. All of VB falls across the resistor, VR=VB. I=VB/R Inductor acts like an short circuit.
Let’s take a special case of no current initially flowing through the inductor Initial charge on capacitor Solution A: B:
I V2 V1 Foolproof sign convention for two terminal devices Label current going in one terminal (your choice). Define voltage to be potential at that terminal wrt the other terminal V= V2 -V1 3. Then no minus signs Power to device KVL Loop Contribution to voltage sum = +V
